16 BAIRE'S THEOREM AND ITS CONSEQUENCES 81
finite subsets of E, but taking E with the weak topology, so that this topology is
defined by the seminorms pXty(U)= \(U • x\y)\ for all x,y in E. This topology is
coarser than the strong topology.
(a) Show that the weak topology is not metrizable (same method as in Problem 8(a)).
(b) The unit ball ^ in &(E; E) is metrizable and compact in the weak topology
(method of (12.5.7)).
(c) The bounded sets in &(E; E) are the same for the weak topology as for the strong
(d) The mapping U\~*U* of &(E; E) into itself is continuous with respect to the
(e) Show that the mapping (U, V) -* UV of V x <$ into tf is not continuous for the
weak topology. (If fe)nez is a Hilbert basis of E indexed by Z, consider the "shift
operator" U defined by U- en — en+1 for all n eZ, and its powers U* and U~n.)
The topology induced on the unitary group U(E) by the weak topology is identical
with that induced by the strong topology, and hence is compatible with the group
structure. Also U(E) is not closed in f for the weak topology.
(f) Show that, for each U0 e &(E; E), the linear mappings Fh-» U0 V and V\-+ VUQ
are continuous on &(E; E) for the weak topology.
(g) Let Jt be a submonoid of *€ (i.e., Jf is stable under multiplication). Show
that the closure of Jt with respect to the weak topology on JS?(E; E) is a compact
monoid, which is commutative if Jt is commutative.
10. The notation is the same as in Problem 9.
(a) Let UeV. Show that the relations U - x = jc, (*| U • x) = (x | *), and U* - x = x
(b) Deduce that the idempotent operators in V are the orthogonal projections
(Sections 6.3 and 11.5.)
11. Let E be a separable Hilbert space, Jt a submonoid containing the identity 1E
which is weakly closed (and therefore weakly compact) in *$ (notation of Problem 8).
For each x e E, the orbit of x under Jt is the set Jt - x of vectors U - x where U e Jt\
it is weakly compact in E. We say that x is a flight vector with respect to Jt if
0 e Jt - x, and that x is a reversible vector with respect to Jf if for each Ue Jt there
exists Ve Jit such that VV • x = x. Let F(^f) (resp. R(uO) denote the set of flight
(resp. reversible) vectors with respect to Jt. If x is reversible, then ||£/*x|| = \\x\\
for all UzJt.
(a) Every orbit Jt • x contains a minimal orbit N (Section 12.10, Problem 6), and
every y e N is reversible with respect to Jt. Let U e Jt be such that U • x = y e N,
and let <stf be the weakly closed submonoid generated by 1E and U: this submonoid
jtf is commutative. Let pc^-^cN be a minimal orbit with respect to jtf,
and let Fe jaf be such that V- y = z e P. Show that there exists We jtf such that
VUW - z = z; deduce that the vector x — W - z is a flight vector, and hence that every
jc e E is the sum of a flight vector and a reversible vector belonging to Jt • x.
(b) Show that if x e E is reversible with respect to Jt then x is also reversible with
respect to every weakly closed submonoid ^T, (Split up x into the sum of a vector
y e R( Jf) belonging to JT • x and a vector z e F(^); use the fact that if a, b are two
distinct vectors in E with the same norm, then ||i(*M~ b)\\ < \\a\\ = \\b\\, and deduce
that y = x.) Deduce that if x e R(^), then given any U e Jt there exists an element
V in the weakly closed submonoid of Jt generated by 1E and U\ such that
VU-x^x. group of a right Cauchy